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where A = 2 iAi. Equation (XXIII) is the integral equation of harmonic motion with the additional factor e~ai, which is called the damping factor. On account of this factor the amplitude of the motion continually diminishes.
It is evident from equation (XXIII) that the motion is periodic and has a period
27T
(XXIV)
The character of the motion is brought out clearly by the displacement-time curve of Fig. 146.   A mental picture of the
FIG. 146.
damped harmonic motion of a particle may be formed by considering the motion of an auxiliary particle which moves in a logarithmic spiral. If the auxiliary particle describes the logarithmic spiral of the figure in the counter-clockwise direction, in such a way as to give the radius vector a constant angular velocity, then the motion of the projection of the auxiliary particle upon the 0-axis is damped harmonic.
The logarithmic spiral may be used as an auxiliary curve in drawing the graph of equation (XXIII), as the circle is used in drawing a sine curve.